27. The diagonal of a polygon is a line which joins the vertices of two angles that are not adjacent, as AC or AD.

29. The circumference of the circle is the bounding curve. 30. An arc is any part of the circumference, as BAF.

31. A chord is a line joining the extremities of an arc, as BF.

32. A diameter is a chord passing through the centre, as DF. 33. A radius is the distance from the centre to the circumference, as CB, CD, CE, etc.

34. A segment of a circle is the portion lying between an arc and its chord, as BAF.

NOTE.-When the chord is diameter, the arc is semicircumference, and the segment is semicircle.

35. A sector of a circle is the portion lying between two radii and their included arc, as BCD or DCE.

The two fixed points are called the foci of the ellipse, as H

and I.

The constant quantity is equal to the longest diameter of the ellipse, viz: AB, the diameter which passes through the foci; thus, HD+ DI= HF+FI= AB.

37. The transverse axis is the longest diameter, AB.

38. The conjugate axis is the shortest diameter, viz: ED, the diameter which is perpendicular to the transverse axis.

39. A tangent line is a line that touches a circle, ellipse, or other curve in one point, and which cannot touch it in any other point, however far the line may be produced or extended in either direction, as GE in the circle, Def. 28, and KL or KM in the ellipse.

40. A polygon is said to be inscribed in a circle or ellipse when the vertex of each angle is in the bounding curve, as ABCD, etc., Def. 26.

The circle or ellipse is then said to be circumscribed about the polygon.

41. A polygon is said to be circumscribed about a circle or ellipse, when each side of the polygon is tangent to the curve, as ABCD, Fig. 10, p. 222.

The circle or ellipse is then said to be inscribed in the polygon.

NOTE.-Mathematical points, lines, and surfaces, exist only in imagination, but we use representations of them to aid us in mathematical investigations.

42. A solid or body is a figure which has length, breadth, and thickness.

43. A prism is a solid that has two similar, equal, parallel faces, called bases, and all its other faces parallelograms.

NOTE. A prism is triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrangles, pentagons, etc.

44. A parallelopipedon is a prism bounded by six parallelograms.

If these six parallelograms are rectangles, the parallelopipedon is rectangular; if they are equal rectangles, it is a cube.

45. A pyramid is a solid, having a polygonal face, called the base, and all its other faces are triangles which meet at a common point, called the vertex of the pyramid.

46. A right pyramid is one whose base is a regular polygon, and in which the perpendicular let fall from the vertex to the base, passes through the centre of the base.

47. An oblique pyramid is one in which the perpendicular, from the vertex to the base, does not pass through the centre of the base.

NOTE.-A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrangle, etc.

48. A cylinder is a round body whose diameter is the same throughout its entire length, and whose ends or bases are equal, parallel circles.

49. A cone is a solid, like a pyramid, except its base is a circle.

NOTE.-The cone is right or oblique in like manner with the pyramid.

50. The frustum of a pyramid or cone is the part remaining after a portion next the vertex has been cut off by a plane parallel to the base.

51. A wedge is a solid bounded by five plane faces, one of which, called the back, is a quadrangle, and usually a rectangle, two of them, called the ends, are triangles, and the other two, called the sides, are trapezoids or parallelograms.

The line in which the sides meet is called the edge of the wedge.

NOTE.-A right wedge has its back and sides rectangles, and .. its ends are parallel to each other, and perpendicular to the back.

52. A rectangular prismoid is a solid resembling the frustum of a rectangular pyramid, but differing from it in this, that the trapezoids forming its sides, if extended, would not meet in a common vertex, but one pair of them would meet before the other, and thus form a wedge instead of a pyramid.

53. A sphere is a solid bounded by a curved surface, all parts of the surface being equally distant from a point within, called the centre.

54. A diameter of the sphere is a line passing through the centre, and limited in both directions by the surface.

NOTE. All diameters of the same sphere are equal.

55. A radius or semidiameter of a sphere is the distance from the centre to the surface.

NOTE 1.-All radii of the same sphere are equal.

NOTE 2.-If a plane be passed through a sphere, the section so made will be a circle. If the plane passes through the centre, the section is a great circle; if it does not pass through the centre, the section is a small circle.

56. A spherical segment is the portion of the sphere cut off by a plane, or the portion lying between two parallel secant or cutting planes.

NOTE 1.-The sections formed by the secant planes are the bases of the segments.

NOTE 2.-A Hemisphere is the segment cut off by a secant plane passing through the centre.

57. A zone is that portion of the surface of a sphere cut off by a secant plane, or the portion included between two parallel secant planes.

58. If two great circles intersect each other upon the surface of a sphere, they bisect each other; i. e. they divide each other into two equal parts; they also have a common diameter.

59. The portion of the surface included between either pair of semicircles is called a lune, and the portion of the sphere cut out by these two semicircles is called an ungula or spherical wedge.

60. If three arcs of great circles enclose a portion of the surface of a sphere, that portion is a spherical triangle; if four or more arcs enclose a portion of the surface, such portion is a spherical polygon.

61. If planes be passed through the arcs of a spherical triangle or polygon, they will pass through the centre of the sphere and form a solid angle at the centre. The portion of the sphere lying between these planes is a spherical pyramid.

62. A spherical sector is a portion of a sphere composed of a spherical segment and a cone having the same base as the segment, and its vertex at the centre of the sphere.

NOTE 1.-The segment, in this case, must have but one base.